So, when I type Japanese, I do so by typing in romaji. So when I type, I see:
Type A ->あ
Type R ->あｒ
Type U ->ある
etc...

When I was watching an anime character type Japanese, they do not type like that. So I was wondering how it is that Japanese people type in Japanese. Do they have a special keyboard with QWERTY and hiragana on the same keys?

There are such keyboards (and in fact I have one, though I find it's not worth the trouble), but in my understanding, very few people use them, even in Japan. I wouldn't really expect to find realistic typing in an anime... you'll notice that computer use on TV and in movies -- in any language -- is rarely realistic either, even when it's live action, let alone in cartoons.

So, most people type exactly the same way you type, though they do have a couple of extra keys to make it a bit easier (to switch from kana to romaji, for example, or hiragana and katakana).

Most Japanese keyboards/laptops will have qwerty and kana on the same keys, but that doesn't mean you have to type on them in kana mode, where one key corresponds to one kana. As furrykef says, most people will type in romaji-mode. Either the character in the anime was typing in kana-mode, or it's just simplistically animated.

That's just that one scene you watched, of course. I've often seen regular Japanese typing in anime. For instance, this section of Azumanga Daioh (jump to 2:15).